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What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

denisegaskins.com

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Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#281

Post by pdonis »

dash2 wrote: post #236
pdonis wrote: post #197

Earlier quoted context omitted.

> multiplication by 1/q Is not repeated addition. You can't add a number to itself 1/q times. At least, not unless you're willing to adopt increasingly perverse interpretations of "repeated addition" as you try to cover more and more numbers. See my response to wruza upthread.

Hmm, so just go about it differently: x * p/q with p,q integers is (x*p) / q. The first bit is repeated addition. The second bit means, find the number r such that r * q = x * p. Even if r is not integer, q is integer, so we can try different numbers, add them to themselves q times, and close in on the answer. So I still think conceptually it's fine to think of it as repeated addition? It might be algorithmically a b…

> The first bit is repeated addition

So now your definition of "repeated addition" is "repeated addition, plus a version of multiplication". Division is the inverse of multiplication, so your definition is circular: you're "defining" multiplication in terms of repeated addition and multiplication.

Similar objections apply to another poster's contention upthread that the "repeated addition" definition is justified because of the distributive law. The distributive law defines A times (b plus c) in terms of A times b plus A times c. So it's useless as a definition of multiplication in terms of addition.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#282

Post by pdonis »

wcarey wrote: post #275
pdonis wrote: post #199

Earlier quoted context omitted.

There's no need to lie to children. Telling children "multiplication is a separate operation on numbers, but it works like repeated addition for the counting numbers you're familiar with" is not a lie.

How would you teach physics without lying to children?

Um, by not lying to them?

Do you have a specific example of where you think you need to lie to teach physics?

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#283

Post by FPGAhacker »

com2kid wrote: post #223
lhorie wrote: post #211

Earlier quoted context omitted.

> teaching students one thing, and then later coming back and saying "well, that thing you were taught before isn't actually correct..." My kids are learning multiplication. I honestly don't feel like they're being lied to, or confused about what they are being taught. Yes, they start with addition and subtraction as crutches (e.g. `9 x N` is initially taught as `10 x N - N`). It's certainly not how I learned (I had…

> (e.g. `9 x N` is initially taught as `10 x N - N`) That isn't a crutch, that is how large numbers can be multiplied in one's head! 90 * 21 is (90 * 20) + 90 Right after removing that extra 90, now you have 9 * 2 * 10 * 10 + 90, which is easy to do mentally. Heck if you ask me what 9 * 14 is, I'm going to do 10 * 14 - 14. Did I have to memorize 0x0 through 12x12? Yup. But now days I have a few key mid points memoriz…

> Heck if you ask me what 9 * 14 is, I'm going to do 10 * 14 - 14.

Yeah, that’s interesting how we do things in our head like that.

I would frequently break that down as 9*10 + 9*4 in my head.

But I might also do it as 10*14 - 10 - 4.

I’ll try different ways in my head until I can solve one of them easily.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#284

Post by pdonis »

SilasX wrote: post #220
pdonis wrote: post #205

Earlier quoted context omitted.

I think this viewpoint is pernicious. A child doesn't have to be "smart" in order to deserve being told the truth. Or I could turn your remark around: what makes you, the adult, think you are so much smarter than the child that you can correctly judge what lies are OK to tell them? Are all adults really that smart? (Are any of us?)

Along with what allturtles said, I don't think it's right to call it a lie, which, to me, implies moving someone's model away from the truth (on the basis of them trusting you to convey the correct one). An oversimplified, "wrong" model doesn't do that; it moves them from ignorance toward the correct model (i.e. increases their prediction accuracy). And yes, even when I'm in the learner's shoes , I prefer that a teac…

> I don't think it's right to call it a lie

It's not a lie to tell them "this is a simplified model that doesn't include everything, but you'll be able to add more complexities to it later". But that's not what "multiplication is repeated addition" says. You would say something like: "repeated addition is a simplified model of multiplication that works for whole numbers, but doesn't work well in more complicated cases that you'll learn about later".

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#285

Post by pdonis »

WalterBright wrote: post #256
pdonis wrote: post #161

Earlier quoted context omitted.

> there is Keith Devlin involved.... I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods while at the same time being taken seriously I don't think his argument (or the similar argument being made in the subject article of this thread) for not teaching students that multiplication is repeated addition is a "pedantic falsehood". I think he has a valid point: that there are downsides…

Teaching Newtonian Mechanics to students is also a lie. But I don't believe starting with General Relativity is going to do anything but confuse students. I was never confused by "3 baskets of 6 apples, how many apples" questions or thought I'd been lied to about addition.

> Teaching Newtonian Mechanics to students is also a lie.

Not if you tell the students that it's an approximate model that works well in the domain they're currently studying, but doesn't work well in a more expanded domain.

Of course if you insist on acting like an authority and telling students that Newtonian Mechanics is "the Truth", then yes, you are lying to them. But you don't have to tell them that to teach them Newtonian Mechanics.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#286

Post by pdonis »

JosephRedfern wrote: post #186
pdonis wrote: post #161

Earlier quoted context omitted.

> there is Keith Devlin involved.... I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods while at the same time being taken seriously I don't think his argument (or the similar argument being made in the subject article of this thread) for not teaching students that multiplication is repeated addition is a "pedantic falsehood". I think he has a valid point: that there are downsides…

I never got over being lied to about electron shells and the 2-8-8 "rule".

How were you lied to about that?

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#287

Post by drdec »

shadowofneptune wrote: post #135
Jtsummers wrote: post #103

Earlier quoted context omitted.

No, it's not. It depends a bit on your teachers but also on which "level" of math you're in (in the US). Lower level but still algebra/geometry classes tend to teach facts, not derivations from foundational concepts. Those are the classes aimed at non-Honors and maybe non-College Prep students (2 of the 3 typical "tracks" students end up in the US, names may vary by state and decade).

The way Geometry is taught in the US is awful. Instead of learning that you can use shapes to do useful calculations like square roots, you slog through postulates and theorums without any sense of why you have to do them. Rarely is what is learnt in geometry ever used in later high school courses, save for trigonometry. I hope it is different in other countries.

The point of the Geometry course in that curriculum is to teach the student how to do proofs. The Geometry itself is secondary.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#288

Post by SilasX »

pdonis wrote: post #284
SilasX wrote: post #220

Earlier quoted context omitted.

Along with what allturtles said, I don't think it's right to call it a lie, which, to me, implies moving someone's model away from the truth (on the basis of them trusting you to convey the correct one). An oversimplified, "wrong" model doesn't do that; it moves them from ignorance toward the correct model (i.e. increases their prediction accuracy). And yes, even when I'm in the learner's shoes , I prefer that a teac…

> I don't think it's right to call it a lie It's not a lie to tell them "this is a simplified model that doesn't include everything, but you'll be able to add more complexities to it later". But that's not what "multiplication is repeated addition" says. You would say something like: "repeated addition is a simplified model of multiplication that works for whole numbers, but doesn't work well in more complicated case…

Yeah, and that's still too much inferential distance to feed to an elementary schooler.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#289

Post by aaplok »

jschwartzi wrote: post #152
cjfd wrote: post #105

Ah yes, there is Keith Devlin involved.... I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods while at the same time being taken seriously. You know, most, if not all, mathematical objects of significance can be defined in many, many different ways. In fact, this is the hallmark of an mathematical object of significance: it keeps popping up in many context and can therefore be def…

Except this isn't a pedantic falsehood. Multiplication has nothing to do with addition and at some point we have to stop teaching students that it's related. Understanding ratio is key to understanding a lot of the physical sciences and you can build a really good intuitive understanding of a lot of simple physical concepts if you can just do dimensional analysis. But if you think of multiplication and division as a…

Multiplication is literally repeated addition, as in "taking a number multiple times". It is in the name.

The fact that the operation is so useful that it has been generalised to the point where that original meaning is eventually lost through more and more abstractions doesn't invalidate that, because with all the generalisation and abstraction, multiplication as repeated addition still works, and any generalisation is expected to leave that property intact.

This is not unique to multiplication. Modern mathematics is all about extracting fundamental properties that make something true, to generalize results while keeping the original simpler statement true. Starting from the generalisations without explaining the intellectual process it took us to get there means just giving facts without explaining them.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#290

Post by pdonis »

lhorie wrote: post #211
pdonis wrote: post #161

Earlier quoted context omitted.

> there is Keith Devlin involved.... I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods while at the same time being taken seriously I don't think his argument (or the similar argument being made in the subject article of this thread) for not teaching students that multiplication is repeated addition is a "pedantic falsehood". I think he has a valid point: that there are downsides…

> teaching students one thing, and then later coming back and saying "well, that thing you were taught before isn't actually correct..." My kids are learning multiplication. I honestly don't feel like they're being lied to, or confused about what they are being taught. Yes, they start with addition and subtraction as crutches (e.g. `9 x N` is initially taught as `10 x N - N`). It's certainly not how I learned (I had…

> I honestly don't feel like they're being lied to, or confused about what they are being taught.

They might not be. If they're just being taught how to multiply in particular cases (whole numbers) using repeated addition, then they're not being lied to. If they're being told straight up that, while multiplication is not identical to repeated addition, they can ease into multiplication by learning repeated addition, even better.

They're only being lied to if they're being told, authoritatively, that multiplication is repeated addition, no ifs, ands, or buts. And yes, I have had teachers like that, and I suspect many others have too.

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